A combinatorial approach to the exponents of Moore spaces

In this article, we give a combinatorial approach to the exponents of the Moore spaces. Our result states that the projection of the $p^{r+1}$-th power map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its atomic piece containing the bottom cell $T^{2n+1}\{p^r\}$ is null homotopic for $n>1$, $p>3$ and $r>1$. This result strengthens the classical result that $\mathrm{\Omega }{T}^{2n+1}\{p^r\}$ has an exponent $p^{r+1}$.